We first illustrate the underlying closure modeling concept for a prototype demonstration as shown in Figure REF .
To formulate our ROM closure problem, we modify Eq. REF  by adding a functional form of distributed control. This control is often referred to as eddy viscosity approach that has strong roots in large eddy simulations to model or compensate the residual effects of the truncated scales [1]}, [2]}, [3]}, [4]}. Therefore, the modified ROM becomes
\( \frac{d \alpha _k}{d t} = \sum _{i=1}^{R} \mathfrak {L}^{i}_{k}\alpha _{i} + \sum _{i=1}^{R} \tilde{\mathfrak {L}}^{i}_{k}\alpha _{i} + \sum _{i=1}^{R}\sum _{j=1}^{R} \mathfrak {N}^{ij}_{k}\alpha _{i}\alpha _{j}, \quad \forall \ k = 1, \ldots , R\) 
